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I'm sure I'm misunderstanding something, but couldn't one measure the extraordinarily large magnetic field that should appear when applying a voltage to (even a tiny sample of) such a material?


A magnetic field is the result of moving an electric charge (in this case electrons). Applying the voltage will move the electrons. The only thing the material being a superconductor changes is that the electrons will lose less energy to resistance while traversing the sample.


No, superconducting states break down in the presence of sufficiently strong magnetic fields.




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